Bearing Capacity Determination Method for Strip Surface Footings Underlain by Voids

Transcription

1 90 TRANSPORTATION RSARCH RCORD 1336 Bearing Capacity Determination Method for Strip Srface Footings Underlain by Voids c. w. HSIH AND M. C. WANG A method of bearing capacity determination for strip srface footings sbjected to vertical central loading and nderlain by a continos circlar void with its axis parallel with the footing axis is presented. A nomograph is also presented for ease in application of the developed eqations. For eqation development, the performance of strip srface footings with and withot an ndergrond void was investigated sing a plane-strain finite element compter program. In the analysis, the fondation soil was characterized as a nonlinear elastic, perfectly plastic material that obeys the Drcker-Prager yield criterion. A wide range of soil properties, footing widths, void sizes, and void locations inclding depth to void and void eccentricity was considered. The ltimate bearing capacity vales were obtained from the reslts of analysis and were related graphically with the inflencing factors investigated. The bearing capacity eqations were then developed throgh crve fitting to these graphical relationships. The effectiveness of the developed eqations was evalated by comparing the compted bearing capacity vales with the model footing test reslts. Good agreement between the two sets of data was shown. It was therefore conclded that the developed eqations together with the nomograph may become an effective tool for analysis and design of strip srface footing nderlain by a continos circlar void, at least within the conditions investigated. Despite its origin, either natrally formed or man-made, an ndergrond void may occr nder a fondation. The freqency of its presence nder the fondation and the seriosness of its effect on fondation stability have been pointed ot in previos papers (1,2). To approach sch a problem, options sch as filling the void, excavating the overlying soil and placing the fondation below the void, sing piles or piers, reinforcing the overlying soil layer, relocating the fondation site, and possibly others are considered. These options are often difficlt and very costly to implement. To ensre its long-term stability, the fondation mst be originally designed with a thorogh nderstanding of the void effect. In response to this need, many research stdies have been condcted (1-9). There is no methodology crrently available for analysis and design of sch a fondation system. The core of the methodology reqires eqations for determination of the ltimate bearing capacity of the fondation. The development of sch bearing capacity eqations for analysis of strip srface footing nderlain by a continos circlar void is presented. C. W. Hsieh, Gannett Fleming, Inc., P.O. Box 1963, Harrisbrg, Pa M. C. Wang, Department of Civil ngineering, The Pennsylvania State University, University Park, Pa FOOTING PRFORMANC ANALYSIS The analysis of strip srface footing performance nder vertical central loading was performed sing the finite element method. A two-dimensional plane-strain finite element compter program was developed. In the compter program, the footing is characterized as a linear elastic material and the fondation soil as a nonlinear elastic, perfectly plastic material that obeys Hooke's law and the Drcker-Prager yield criterion (10). A hypothetical layer to model possible slips at the interface is placed between the vertical sides of the footing and the soil. The nonlinear soltion is accomplished throgh the se of Reyes's incremental stress-strain relation (11) together with the tangential stiffness method. The compter program differs from those previosly developed by Bas (12), Badie (13), and Azam (14) in that it contains a preprocessor, the main program, and a postprocessor. The preprocessor is for inpt data preparation, and the postprocessor is for providing the reslts of finite element analysis in the desired graphical form. The preprocessor adopts a step-by-step procedre sing the qestion-and-answer interactive format to gide the ser to provide the necessary inpt. It contains varios sbrotines for generating the finite element mesh and for specifying bondary conditions, material properties, external loading, and type of analysis. The program also incorporates the nmerical scheme of Siriwardane and Desai (15) for keeping the state of stress dring yielding on the yield srface, and the nmerical comptation ses the Gass-Jordan elimination method to solve the symmetrical BANDD global stiffness matrix. A detailed description of program development is given by Hsieh (16). The compter program was validated by sing the model footing test data obtained by Bas (12), Badie (13), and Azam (14) and the data pblished by Siriwardane and Desai (15) and Whitman and Hoeg (17). The finite element analysis was performed sing an IBM The generation of element mesh and other inpt data preparation was done throgh a VAX 8550 or a PC, and a VAX 8550 was sed to obtain the reslts of analysis. For plotting software, PLOT 10 was sed. FOUNDATION SYSTM INVSTIGATD The strip srface footing analyzed was a reinforced concrete footing with a width varying between 2 in. and 6 ft. Three different soils spport the footing-commercial kaolin, silty clay, and clayey sand. The ndergrond void was circlar in

2 Hsieh and Wang 91 cross section and continos with its axis parallel to the footing axis. Figre 1 shows a schematic view of the footing/soil/ void system together with the symbols sed for defining footing size and the size and location of the void. As shown, B, D,, and W represent footing width, depth to void, void eccentricity, and void diameter, respectively. In the analysis, the size (W) and location (D and ) of the void were expressed as ratios to footing width (B). Three levels each of void size (WIB = 0.67, 1, and 2), void eccentricity (IB = 0, 1, and 2), and nmeros levels of depth to void (DIB = 1to14) were analyzed. The material properties sed in the analysis are smmarized in Table 1. The concrete footing properties in Table 1 are obtained from Bowles (18), and the properties of silty clay, kaolin, and clayey sand are from Bas (12), Badie (13), and Azam (14), respectively. TABL 1 PROPRTIS OF FOUNDATION SOILS AND CONCRT FOOTING USD IN FINIT LMNT ANALYSIS Material Parameters Kaolin Silly Clayey Concrete Clay Sand Footing Internal Friction Angle, NIA deg Unit Cohesion (psi) NIA Initial Modls in x106 Compression (psi) Initial Modls in Tension x106 (psi) Poisson's Ratio Dry Unit Weight, pci ULTIMAT BARING CAPACITY Reslts of the finite element analysis were sed to evalate the mechanistic behavior of the fondation system. Among the behaviors investigated are principal stress distribtion, displacement fields, propagation of plastic yielding, deformed configrations of the void, and the footing pressre verss settlement relation. From these data, the ltimate bearing capacity of each condition analyzed was obtained. The ltimate bearing capacity data were analyzed frther with respect to varios factors considered inclding soil type, footing width, void size, and void location. It was fond that the effect of soil type and footing width on bearing capacity for footing nderlain by a void can be properly considered by nondimensionalizing both the independent and dependent variables: the ratios of q 0 /qnv DIB, WIB, and IB, in which q 0 and qnv are the ltimate bearing capacities of with and withot void conditions, respectively (16). Frthermore, the ltimate bearing capacity vales can be related with void locations for different void sizes in the form shown in Figres 2 throgh 6 regardless of soil type. Note that the data in,., :; c Ol! z 3 "' DIB FIGUR 2 Variation of ltimate bearing capacity with DIB for kaolin with /B = 0 and W/B = 0.67, 1.0, and 2.0. FOOTING LOAD i y--- SOIL _,)., D + w _L -----I FIGUR 1 Schematic view of footing/soil/void system.! 150,.,. :; Ol z D/B FIGUR 3 Variation of ltimate bearing capacity with DIB for kaolin with IB = 1.0 and WIB = 0.67, 1.0, and 2.0.

3 92 TRANSPORTATION RSARCH RCORD (1,2), varies with soil type, void size, and void eccentricity. The critical depth to void is an important factor reqired in the ltimate bearing capacity eqation "o i 100 "' : ::> < D/B FIGUR 4 Variation of ltimate bearing capacity with DIB for kaolin with IB = 2.0 and W!B = 0.67, 1.0, and 2.0. Figres 2, 3, and 4 are for kaolin with IB = 0, 1, and 2, respectively; Figres 5 and 6 are respectively for silty clay and clayey sand with IB = 0. Other figres are available elsewhere (16). Ths, there is a total ofnine figres. These graphs form the data base for the development of the ltimate bearing capacity eqation. According to Figres 2 throgh 6 and the other figres, the ltimate bearing capacity of strip footing varies with void size and location so that the bearing capacity increases as the depth to void increases, void size decreases, and void eccentricity increases, while other factors are constant. The data indicate that there is a depth to void beyond which the presence of a void has no effect on the ltimate bearing capacity. This depth to void, termed as the critical depth (De) in previos papers QUATION FORMULATION The ltimate bearing capacity eqation was developed by fitting the graphical relations in Figres 2 throgh 6 and the other figres. In the fitting process, the main featres of the crves were first identified. Fnctions having sch featres were then selected to fit the crves. Throgh trial and error, the fnction best fitting the crve was adopted. Frther, the coefficient, amplitde, and argment of the fnction were determined for each crve. The varios sets of coefficients for all crves were analyzed to determine the variation between each coefficient and the inflencing factors. For the crves in Figres 2 throgh 6, different fnctions were tried, sch as polynomial, hyperbolic secant, arc tangent, and sine. Of the varios fnctions attempted, the one-qartercycle sine fnction best fit the crves. The fnction contains two coefficients-one for the intercept on the vertical axis and the other for the amplitde. By sing this fnction, the SAS nonlinear regression analysis was performed to determine the coefficients that best fit each crve. It was fond that the argment in the sine fnction eqals ;, and the coefficients vary with the void size, void location, and shear strength property of soil. The eqation relating the coefficients with void size, void location, and soil strength was developed first by plotting the coefficients against WIB for each IB and soil type. All the relation crves were then fitted by a factored hyperbolic secant fnction that has a maximm vale eqal to qnv the ltimate bearing capacity vale of no-void condition, and is asymptotic to a constant vale as WIB approaches infinity. Frthermore, to generalize the eqation that fits the crves in Figres 2 throgh 6, the ltimate bearing capacity is expressed as q)qnv In the analysis, the vale of qnv is compted "o,., - Ii" c 50 - "'! e 25 p D/B FIGUR 5 Variation of ltimate bearing capacity with DIB for silty clay with IB = 0 and WIB = 0.67, 1.0, and 2.0.

4 Hsieh and Wang 93 60! 40 ll. 20 "' :B ::::> i / FIGUR 6 Variation of ltimate bearing capacily with DIB for clayey sand with IB = 0 and WIB = 0.67, 1.0, and 2.0. by sing the conventional bearing capacity eqation together with Meyerhof's coefficients. Becase the argment of the one-qarter-cycle sine fnction contain the critical depth to void (De) an eqation for predicting D i reqired. From Figres 2 throgh 6 and the others the critical depth to void for each condition i obtained from the point where the crve reaches the no-void bearing capacity vale. Tbe vales of De ths obtained are plotted again t WIB for each level of IB. The plots reveal that the Dc!B verss log (WIB) relation can be approximated by a linear fnction. The coefficients contained in the linear fnction of DJB verss log (WIB) are then frther related with void location and soil strength property. Details on eqation formlation are docmented elsewhere (16). The final ltimate bearing capacity eqation is as follows, where ' is the bearing capacity ratio: In qation 1, ' = sin B + K(l - sin B) and (1) (2) In the preceding eqations, <!> = internal friction angle of soil, c = cohesion of soil, and -y = nit weight of soil. NOMOGRAPH AND XAMPL For ease in application of the developed eqations one nomograph for determination of K is presented in Figre 7. There are three sets of crves-figres 7a, b, and c for internal friction angle (<I>) eqal to 10, 20 and 30 degees, respectively. A minimm of three levels for each variable is al o presented in the nomograph for ease in data interpolation when necessary. A an example of the se of the nomograph and eqations, a 5-ft-wide strip srface footing is pported by cohesive soil. The fondalion oil has a nit weight, cohesion, and internal friction angl.e of 130 pcf, 15 p i, and 20 degrees, respectively. A 5-ft diameter continos circlar void having its axis parallel with the footing axis is 3 ft from the footing axis and 10 ft below the footing base. The ltimate bearing capacity of the footing can be determined a follows. For the conditions given, B = 5 ft, = 3 ft, W = 5 ft, and D = 10 ft, -y = 130 pcf, c = 15 psi, and<!> = 20 degrees. Dc = B[D 1 + D 2 log (WIB)] D 1 = 16.3sin 2<1> cos 2 <1> (/B) sin 2<1> D 2 = 22.5sin <!> - 3.5(IB) tan <!> K =KP+ sech{[2.9 - tan 2 <1> - 0.4(/B)2cos2<!>] + [ tan 2 <1> (/B)cos 2 <!>]Iog(W/B)} K = {-0.42 tan 2 <1> if 2cB -yb 2 p 0 if 2cB < -yb 2 (3) (4) (5) (6) (7) 1. Determine D. and compte B. From qations 4, 5, and 6, for IB = 0.6, WIB = 1, and <I> = 20 degrees, De= 44 ft B = D = = DC Determine coefficient K. From Figre 7(b), for WIB = 1, IB = 0.6, <I> == 20 degrees, and 2cB = 2 (15 x 144)(5) = 21600, which is greater than -yb 2 = (130)(5) 2 = 3250, K = 0.08

5 94 TRANSPORTATION RSARCH RCORD <1>= 10 when 2Bc yb 1!>--& /B 2 -B--& /B I "'t: -7 6-/B O ' when 2Bc < yb 2._._.,. /B 2 N --/B I -/B O I -:: I r=:::. t::::.:: Compte bearing capacity ratio ' ' sin 8 + K (l - sin 8) sin (1 sin ) = Compte the ltimate bearing capacity of no-void condition, qnv For <I> = 20 degrees, Ne = 14.83, Nq = 6.40, and N'Y = 2.90, (a) W/B = (15)(14.83) + o + (i2 )c5 x 12)(2.90) = psi 5. Compte the reqired ltimate bearing capacity <;6=20 when 2Bc:!:yB _,_-& 2 /B 2 -e--e- / B= 1 -e--e-/b=o when 2Bc<yB - 2../8 2 --/B=I "- -- -/B=O !'-, ::"""- -s -- I'""- --, --f=::::: g --.,. r:::: f:::i I}--: J--, r---=::::::::--:: _ (b) (c) W/ B <1>=30 when 2 B c y 8 1 -A--<>-/8 2 -e,-s-/b=l -&-6-/B O when 2 B c < y 8 2 /B 2 --/B=I -/B O , -- -e _ -.,, :.::: ::::; =:::: W/B FIGUR 7 Variation of coefficient K with W/B for internal friction angle of (a) 10, (b) 20, and (c) 30 degrees. (0.40)(229.0) 91.6 psi COMPARISONS To demonstrate the effectiveness of the developed eqations, the ltimate bearing capacity of model test footings was compted and compared with the test reslts obtained by Badie (13) for kaolin and Azam (14) for clayey sand. The model test footing was a steel plate 2.0 in. wide by 5.25 in. long by 0.5 in. thick, tested nder the plane-strain loading in a test tank approximately 32 in. high, 60 in. long, and 5.5 in. wide. A circlar void was at varios locations. The conditions, inclding void location and soil type sed for comparison, test reslts, and compted bearing capacity vales, are given in Table 2. A fairly good agreement between the two sets of data is seen, indicating that the developed eqations can provide accrate bearing capacity data for strip srface footing nderlain by a continos void at least within the range of conditions investigated. SUMMARY AND CONCLUSIONS The need to develop a methodology for determining the ltimate bearing capacity of shallow fondation nderlain by TABL 2 COMPARISON BTWN COMPUTD ULTIMAT BARING CAPACITY AND MODL FOOTING TST RSULT Ultimate Bearing Capacity (psi) Test Soil D/B WIB /B Compte<! Test Reslt Kaolin Clayey Sand

6 Hsieh and Wa11g an ndergrond void was identified. A method of bearing capacity determination for strip srface footing overlying a continos circlar void with its axis parallel to the footing axis was presented. For bearing capacity eqation development, the performance of strip srface footing sbjected to a vertical central loading with and withot an ndergrond void was investigated sing a plane-strain finite element compter program. In the analysis, the fondation soil was characterized as a nonlinear elastic, perfectly plastic material that obeys the Drcker-Prager yield criterion. To cover a wide range of soil property, three different soils were analyzed-kaolin, silty clay, and clayey sand. A range of footing width, varying void sizes, and void locations inclding the depth to void and void eccentricity were considered. The ltimate bearing capacity of each condition analyzed was obtained from the footing performance data. These ltimate bearing capacity vales were then related graphically with the varios inflencing factors investigated inclding void size, void location, and shear strength properties of the soil. The bearing capacity eqations were developed throgh crve fitting of the graphical relationships. For these eqations, one nomograph was presented for ease in eqation application. The developed eqations were sed to determine the bearing capacity of some model test footings, and the reslts were compared with the test data. A good agreement between the prediction and test data was obtained. On the basis of this comparison, it may be conclded that the developed bearing capacity eqation may become an effective tool for analysis and design of strip srface footing nderlain by a circlar void, at least within the range of conditions considered. ACKNOWLDGMNTS The athors are gratefl to the National Science Fondation for its financial spport of the research reported in this paper. The athors thank Karen M. Detwiler for painstakingly typing the manscript. RFRNCS 1. R. L. Bas and M. C. Wang. TI1c Bearing Capaci ty of Strip Footing Located A bove a Void in Cohesive Soils. Jomal of Geotechnical 11gi11eeri11g Division, ASC, Vol. 109, No. J, 1983, pp A. Badie and M. C. Wang. Stability of Spread Footing Above Void in Clay. Jornal of Geotechnical ngineering Division ASC, Vol. 110, No. 11, 1984, pp M.. Wang and R. L. Bas. elllement of Footing Above a Void. 2nd Conference 011 Gro 1111d Movement and Strctres, The Dcparlm nt of Civil ngineering and Bilding Technology UWIST, M. C. Wang and A. Badie. ffect of Undergrond Vo id on Fondation Stability. Jornal of Geotechnica/ 11gineeri11g, ASC, Vol. 111, No. 1985, pp M. C. Wang, C. S. Yoo, and C. W. Hsieh. ffect of Void on Footing Behavior nder ccentric and Inclined Loads. Fondation ngineering: Crrent Principles mid Practices, Vol. 2 Proc. of the Congress, ASC, L989 pp L.A. W od and W. J. Larn ach. T he Behavior of Fooring Located Above Voids. Proc., leventh lntematio11a/ Conference 011 Soil Mechanics and Fo11nda1ion ngineering, Vol. 4, San F.ranci co, Calif., pp G. A. H. Abdellah and M. H. Abdalla. The Interaction Between a Tnnel/Cavity and Nearby Strctres. Proc., VI Astralian Tnnelling Conference, 1987, Vol. I. pp. l M. C. Wang and C. W. Hsieh. Collapse Load of SLrip Footing Above Circlar Void. Jo11mal of Geoteclmical ngineering, ASC, Vol. 113 No. 5, 1987, pp. Sll A. Badie and M.. Wang. Stability of Undergrond avity Sbjected 10 rface Load. lntemationa/ Symposim on U11iqe Undergro1111d S1rct11res, Denver, olo D. Drcker and W. Prager. Soil Mechanics and Plastic Analysis in Limit Design. Q arterly of Applied Mathemalics, Vol. 10, No. 2, 1952 pp S. F. Re y. la tic-plas1ic A11alysis of Undergrond Openings by the Finite lement Melhod. Ph.D. thesis. University of Illinois, Urbana R. L. Bas. The S111bility of Shallow Continos Footings Located Above Voids. Ph.D. thesis. The Pennsylvania State University, University Park A. Badie. Stabili1y of Spread Foo1ing Spported by Clay Soil with Undergro1111d Void. Ph.D. thesis. The Pe nnsylvania State University, University Park, G. Azam. Swbility of Shallow Co111i s Footings S11pported by Two-Layer Soil Deposits ll'ith a11 Undergrond Void. Ph.D. Lhcsis. The Pennsylvania State Univer ity. Uni vc r ity Park, H. J. Siriward ane and.. De ai. omptational Procedres r r Nonlinear Three-Dimensional Analysis with Some Advanced Constittive Law. lmemmio11a/ Jornal for N11merical and Analytical Me1hods in Geomechanics, Vol. 7 No. 2, pp C. W. Hsieh. Development of Metliodology for Stability A11alysis of Srface Strip Footing Above Co111i11o11s Circlar Void. Ph.D. dissertation. Department of Civil ngineering, The Pennsylvania State Universit y, University Park, R. V. Whitman and K. Hoeg. Development of Plastic Zone Beneath a Footing. Report 10 U.. Army 11gl11eeri11g Waterways p e rim em Station Department of Civil ngineering. T he Massachsett In titte of Technology, Cambridge, J.. Bowles. Fowlation Analysis and Design (4th ed.). McGraw Hill Book Co., New York, Pblication of this paper sponsored by Commillee on Fondations of Bridges and Other Strctres. 95